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Can someone please help me with some integral homework before I shoot the crap out of my calculus book pretty please?! Equation in the forum

Mathematics
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\[\int\limits_{}^{}\ln (\sqrt[3]{x})\] That is x to the 3 root
hi there
hello = )

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Other answers:

it might help to think of the cube root of x as x^(1/3)
then use the power rule
ok do u want me to solve it or should i tell u how to do it
ok I got as far to as thinking of x^(1/3) lol
now use the rule that integral of x^n = (1/(n+1))*x^(n+1)
if u solve it i give u medal and if i do so you give me one
so do u mean put 1/3 in front of the ln of stuff and then put it in front of the entire integral? bc that's what I did and it said it was wrong = (
there's no ln involved here
u=x^1/3
ok maybe you should write the whole thing out cuz now you're just confusing me, please, thank you
does this rule look familiar |dw:1360297226043:dw|
sry the +C is nearly cut off
no yeah i already know that lol we're in the chapter of integration by parts and there is a ln involved it's in the front of the x^(1/3)
not to interrupt, but i think the starting point here is \[\frac{1}{3}\int \ln(x)dx\] which you do either by parts or by memory
oh my bad, i didn't see the ln at the very beginning just thought it was cube root of x
its ok
since \(\ln(x)\) is a very common function, it might be benificial just to memorize \[\int \ln(x)dx=x\ln(x)-x\] you can check by differentiation
yes that's what i did. oh damn nvm hahaha i think i get it I keep getting it confused that the derivative of ln is 1/x and we don't know the integral of ln yet so we have to do integration by parts lol
your answer is therefore \[\frac{1}{3}\left(x\ln(x)-x \right)\]
you can use taylor series to integrate this easily
but with z=x+1
you guys were a great help! ty! = )
i would memorize it just like remembering that \[\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}\]so while your fellow classmates are integrating by parts, you just write down the answer
i won't recommend that method
what about\[\int\limits \ln e^{^{x ^{2}}}dx\]
or
\[\int\limits \ln(x^x)dx\]
or
\[\int\limits \ln(x^2+2x+1)dx\]
so these are 3 examples that can't be used by just memorizing the specific case
so basically use integration by parts like ur class mates

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